Born in Cooper Street, Manchester, the oldest daughter of Hannibal Becker, whose father, Ernst Becker had emigrated from Ohrdruf in Thuringia. Becker was educated at home, like many girls at the time. Intellectually curious, she studied botany and astronomy from the 1850s onwards, winning a gold medal for an 1862 scholarly paper on horticulture. An uncle, rather than her parents, encouraged this interest. Five years later, she founded the Ladies' Literary Society in Manchester. She began a correspondence with Charles Darwin and soon afterwards convinced him to send a paper to the society. In the course of their correspondence, Becker sent a number of plant samples to Darwin from the fields surrounding Manchester. She also forwarded Darwin a copy of her "little book", Botany for Novices (1864). Becker is one of a number of 19th-century women who contributed, often routinely, to Darwin's scientific work. Her correspondence and work alike suggest that Becker had a particular interest in bisexual and hermaphroditic plants which, perhaps, offered her powerful 'natural' evidence of radical, alternative sexual and social order.
She was also recognised for her own scientific contributions, being awarded a national prize in the 1860s for a collection of dried plants prepared using a method that she had devised so that they retained their original colours. She gave a botanical paper to the 1869 meeting of the British Association for the Advancement of Science about the effect of fungal infection on sexual development in a plant species Botany remained important to her, but her work for women's suffrage took over the central role in her life. Her involvement in promoting and encouraging scientific education for girls and women brought these two aspects together.
In autumn 1866 Becker attended the annual meeting of the National Association for the Advancement of Social Science, where she was excited by a paper from Barbara Bodichon entitled "Reasons for the Enfranchisement of Women". She dedicated herself to organising around the issue, and in January 1867 convened the first meeting of the Manchester Women's Suffrage Committee, the first organisation of its kind in England.
Several months later, a widowed shop owner, Lilly Maxwell, mistakenly appeared on the register of voters in Manchester. She was not the first but she was a good opportunity for publicity. Becker visited Maxwell and escorted her to the polling station. The returning officer found Maxwell's name on the list and allowed her to vote. Becker immediately began encouraging other women heads of households in the region to petition for their names to appear on the rolls. Their claims were presented in court by Sir John Coleridge and Richard Pankhurst in Chorlton v. Lings, but the case was dismissed.
On 14 April 1868, the first public meeting of the National Society for Women's Suffrage in the Free Trade Hall in Manchester. The three main speakers were Agnes Pochin, Anne Robinson and Becker. Becker moved the resolution that women should be granted voting rights on the same terms as men.